We study the gradient flow associated with the functional $F_\phi(u)$
:= $\frac{1}{2}\int_{I} \phi(u_x)~dx$, where $\phi$ is non
convex, and with its singular perturbation
$F_\phi^\varepsilon(u)$:=$\frac{1}{2}\int_I (\varepsilon^2
(u_{x x})^2 + \phi(u_x))dx$. We discuss, with the support of numerical
simulations, various aspects of the global dynamics of solutions
$u^\varepsilon$ of the singularly perturbed equation $u_t = - \varepsilon^2
u_{x x x x} + \frac{1}{2}
\phi''(u_x)u_{x x}$ for small
values of $\varepsilon>0$. Our analysis
leads to a reinterpretation
of the unperturbed equation $u_t = \frac{1}{2} (\phi'(u_x))_x$,
and to a well defined notion of a solution. We also examine the
conjecture that this solution coincides with the limit of $u^\varepsilon$
as $\varepsilon\to 0^+$.